The local analysis of point singularities of the two dimensional Willmore surfaces is motivated by the notion of concentration compactness: sequences of Willmore immersions with uniformly bounded energy converges everywhere except at finitely many points where the Willmore energy is concentrated. Such point singularities, which naturally occur as blow-ups of the Willmore flow, were first analytically investigated in codimension one by Kuwert and Schätzle (see , ).
In [1]}, they proved the regularity of Willmore surfaces having unit density singularity; and later considered a more general case of Willmore surfaces with higher order singularity [2]}. Their results were generalised to higher codimension by Bernard and Rivière in [3]} via the reformulation of the Willmore equation in divergence form and techniques of integration by compensation (see also [4]}).
